Huber Theorem revisited in dimensions 2 and 4

TL;DR

This paper revisits the second Huber theorem in dimensions 2 and 4, establishing new regularity results for metrics with singularities using Coulomb frames and conformal transformations.

Contribution

It introduces novel methods for handling singularities in metrics in dimensions 2 and 4, including Coulomb frame techniques and regularity conditions for Bach-flat metrics.

Findings

Proves a new version of the 2D Huber theorem with Sobolev curvature assumptions.

Constructs regular conformal metrics across singularities in 4D with $L^p$-bounded Bach tensor.

Provides a framework for analyzing singularities in Bach-flat metrics and immersions.

Abstract

We study the second Huber theorem in dimensions 2 and 4. In dimension 2, we prove a new version assuming that the Gauss curvature lies in a negative Sobolev space using Coulomb frames. In dimension $4$, given a metric having a pointwise singularity with $L^p$-bounds on the Bach tensor, we construct a conformal metric which is regular across the singularity. To do so, we introduce another Coulomb-type condition, similar to the case of Yang--Mills connections. This enables us to obtain a conformal metric satisfying an $\varepsilon$-regularity property. We obtain a generalization of the two-dimensional case that can be applied to study the singularities of Bach-flat metrics and immersions with second fundamental forms in $W^{2,\frac{4}{3}+\varepsilon}$.